Indeterminate Forms in Limits
When working with limits, you may come across an indeterminate form - a situation where the result isn’t immediately clear and further analysis is required. These are the most common indeterminate forms in limit problems:
Just because a limit results in an indeterminate form doesn’t mean the limit doesn’t exist.
To evaluate it, the expression must be rewritten or simplified in a way that removes the source of indeterminacy.
How to Resolve an Indeterminate Form
There are several strategies you can use to evaluate limits that take on an indeterminate form:
- Algebraic manipulation. Sometimes, a limit becomes solvable after applying a few algebraic transformations. Rewriting the expression in an equivalent form can often eliminate the ambiguity. However, this approach isn’t always straightforward, and a suitable transformation may not always exist.
Example 1. This limit presents an indeterminate form of type 0/0: $$ \lim_{n \rightarrow \infty} \ \frac{\frac{1}{n^2}}{\frac{1}{n}} = \frac{0}{0} $$ By simplifying the expression algebraically, we can resolve the indeterminacy: $$ \lim_{n \rightarrow \infty} \frac{1}{n^2} \cdot \frac{n}{1} = \lim_{n \rightarrow \infty} \frac{1}{n} = 0 $$ Example 2. In some cases, exponential and trigonometric functions can be approximated using Taylor series expansions to evaluate the limit.
- L’Hôpital’s Rule. When a limit takes on the form 0/0 or ∞/∞ and the functions involved are differentiable, you can apply L’Hôpital’s Rule. This involves taking the derivative of the numerator and denominator - repeating if necessary - until the limit becomes determinate (finite or infinite).
Example. Consider the following limit, which initially appears as ∞/∞: $$ \lim_{n \rightarrow \infty} \frac{n-1}{n} = \frac{\infty}{\infty} $$ Applying L’Hôpital’s Rule by differentiating numerator and denominator: $$ \lim_{n \rightarrow \infty} \frac{D[n-1]}{D[n]} = \lim_{n \rightarrow \infty} \frac{1}{1} = 1 $$ Therefore, the limit evaluates to 1.
Determinate Forms
Not every occurrence of zero or infinity in a limit leads to an indeterminate form.
For example, the following expressions are not indeterminate:
| $$ k + \infty = +\infty $$ | A real number k added to infinity |
| $$ k - \infty = -\infty $$ | A real number k minus infinity |
| $$ \infty+\infty = \infty $$ | Infinity plus infinity |
| $$ k \cdot \infty = \infty $$ | A nonzero constant k multiplied by infinity |
| $$ \infty \cdot \infty = \infty $$ | Infinity times infinity |
| $$ \frac{k}{\infty} = 0 $$ | A constant k divided by infinity |
| $$ \frac{0}{\infty} = 0 $$ | Zero divided by infinity |
| $$ \frac{\infty}{k} = \infty $$ | Infinity divided by a nonzero constant k |
| $$ \frac{k}{0} = \infty $$ | A nonzero constant k divided by zero |
In these cases, the outcome is algebraically clear and well-defined. They are not considered indeterminate forms.
And so on.
